To resolve this question, we need to analyze both the assertion and the reason, and determine whether both statements are true and whether the reason correctly explains the assertion.
Assertion $A$: Butylated hydroxyanisole (BHA) when added to butter increases its shelf life.
Explanation for $A$: BHA is a well-known antioxidant. Antioxidants prevent the oxidation of fats in foods, such as butter, which helps in retarding rancidity. Rancidity is a process that leads to the spoilage of food. By inhibiting oxidation, BHA effectively extends the shelf life of the butter. Therefore, the assertion is correct.
Reason $R$: Butylated hydroxyanisole is more reactive towards oxygen than food.
Explanation for $R$: Antioxidants like BHA act as sacrificial agents. They react with oxygen before it can react with food products, thus protecting the food from oxidative damage. This statement highlights the fundamental property of antioxidants, hence it is correct.
Thus, both Assertion $A$ and Reason $R$ are correct.
Linking A and R: Since BHA reacts with oxygen more readily than food does, it prevents oxidation of the food and thereby increases the shelf life. Thus, the reason provided ($R$) correctly explains why the assertion ($A$) is true.
Considering the analysis above, the most appropriate answer is:
Both $A$ and $R$ are correct and $R$ is the correct explanation of $A$.